Refined Eigenvalue Bounds on the Dirichlet Fractional Laplacian

Abstract

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta {)}^{\alpha /2}{|}_{\Omega }$ restricted to a bounded domain $\Omega \subset {\mathbb{R}}^d$ with $d=2,$ $1\le \alpha \le 2$ and $d\ge 3,$ $0<\alpha \le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].

14 pages, 2 figures

References in corpus (2)

• Two dimensional Berezin-Li-Yau inequalities with a correction term
• A refinement of the Berezin-Li-Yau type inequality for nonlocal elliptic operators